Problem 13 · 1997 AJHSME
Hard
Fractions, Decimals & Percents
weighted-average
Three bags of jelly beans contain 26, 28, and 30 beans. The fractions of yellow beans in the bags are 50%, 25%, and 20%, respectively. All three bags are poured into one bowl. Which of the following is closest to the percent of yellow beans in the bowl?
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Answer: A — About 31%.
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Hint 1 of 2
You can't just average 50%, 25%, 20% — the bags hold different amounts. Go back to actual beans: count the yellows, count the total, then take the overall ratio.
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Hint 2 of 2
A combined percentage is a weighted average: total of the parts ÷ total of the wholes, never the plain mean of the rates.
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Approach: real counts: yellow total over grand total
- Convert each rate to a count, choosing the easy fraction: 50% of 26 = 13, 25% of 28 = 7, 20% of 30 = 6.
- Yellow total = 13 + 7 + 6 = 26; bean total = 26 + 28 + 30 = 84. The bowl's ratio is 26/84 ≈ 30.9%, closest to 31%.
- Trap: averaging the percentages, (50+25+20)/3 ≈ 31.7%, lands near the same answer here by luck — but that shortcut is wrong whenever the bag sizes differ much. Always weight by how many beans each bag has.
- You'll see it again: mixing groups of different sizes always calls for a weighted average — sum the actual amounts, then divide.
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